Homology, Homotopy and Applications

Volume 10 (2008)

Number 1

The Euler characteristic of a category as the sum of a divergent series

Pages: 41 – 51

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n1.a3


Clemens Berger (Laboratoire J.-A. Dieudonné, Université de Nice, Nice, France)

Tom Leinster (Department of Mathematics, University of Glasgow, Scotland, United Kingdom)


The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one.


Euler characteristic, finite category, divergent series, divergent su, Möbius inversion

2010 Mathematics Subject Classification

05C50, 18Fxx, 40A05, 57N65

Published 1 January 2008